Optimal. Leaf size=55 \[ \frac {x \left (\frac {d x}{c}+1\right )^{-n} \left (c x^2+d x^3\right )^n \, _2F_1\left (-n,2 n+1;2 (n+1);-\frac {d x}{c}\right )}{2 n+1} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2011, 66, 64} \[ \frac {x \left (\frac {d x}{c}+1\right )^{-n} \left (c x^2+d x^3\right )^n \text {Hypergeometric2F1}\left (-n,2 n+1,2 (n+1),-\frac {d x}{c}\right )}{2 n+1} \]
Antiderivative was successfully verified.
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Rule 64
Rule 66
Rule 2011
Rubi steps
\begin {align*} \int \left (c x^2+d x^3\right )^n \, dx &=\left (x^{-2 n} (c+d x)^{-n} \left (c x^2+d x^3\right )^n\right ) \int x^{2 n} (c+d x)^n \, dx\\ &=\left (x^{-2 n} \left (1+\frac {d x}{c}\right )^{-n} \left (c x^2+d x^3\right )^n\right ) \int x^{2 n} \left (1+\frac {d x}{c}\right )^n \, dx\\ &=\frac {x \left (1+\frac {d x}{c}\right )^{-n} \left (c x^2+d x^3\right )^n \, _2F_1\left (-n,1+2 n;2 (1+n);-\frac {d x}{c}\right )}{1+2 n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 0.96 \[ \frac {x \left (x^2 (c+d x)\right )^n \left (\frac {d x}{c}+1\right )^{-n} \, _2F_1\left (-n,2 n+1;2 n+2;-\frac {d x}{c}\right )}{2 n+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x^{3} + c x^{2}\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{3} + c x^{2}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{3}+c \,x^{2}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{3} + c x^{2}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 56, normalized size = 1.02 \[ \frac {x\,{\left (d\,x^3+c\,x^2\right )}^n\,{{}}_2{\mathrm {F}}_1\left (2\,n+1,-n;\ 2\,n+2;\ -\frac {d\,x}{c}\right )}{\left (2\,n+1\right )\,{\left (\frac {d\,x}{c}+1\right )}^n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c x^{2} + d x^{3}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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